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The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]